On the total length of external branches for beta-coalescents

Abstract

In this paper we consider the beta(2 - α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch length L_ext⁽ⁿ⁾ of an initial sample of n individuals. For this class of coalescents, it has been proved that n^α-1T⁽ⁿ⁾ →^D T, where T⁽ⁿ⁾ is the length of an external branch chosen at random and T is a known nonnegative random variable. For beta(2 - α, α)-coalescents with 1 < α < 2, we obtain lim_n→+∞n^3α-5 E{(L_ext⁽ⁿ⁾ - n^2-αE{T})²} = ((α - 1)Γ(α + 1))²Γ(4 - α) / ((3 - α)Γ(4 - 2α)).